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37
Symmetry reductions and exact solutions of a class of nonlinear heat equations
 Physica D
, 1993
"... Classical and nonclassical symmetries of the nonlinear heat equation ut = uxx + f(u), (1) are considered. The method of differential Gröbner bases is used both to find the conditions on f(u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to sol ..."
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Cited by 57 (4 self)
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Classical and nonclassical symmetries of the nonlinear heat equation ut = uxx + f(u), (1) are considered. The method of differential Gröbner bases is used both to find the conditions on f(u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic f(u) in terms of the roots of f(u) =0. 0 Symmetry Reductions of a Nonlinear Heat Equation 1
Review of Symbolic Software for the Computation of Lie Symmetries of Differential Equations
 Euromath Bull
, 1999
"... A survey of symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are briey outlined. Examples illustrate the use of the software. Directions for further research and development are ..."
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Cited by 49 (3 self)
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A survey of symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are briey outlined. Examples illustrate the use of the software. Directions for further research and development are indicated.
Algorithms for the nonclassical method of symmetry reductions
 SIAM Journal on Applied Mathematics
"... ar ..."
Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems
 Adv. Math
, 2000
"... Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinar ..."
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Cited by 11 (3 self)
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Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasiexactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented.
Evolution equations, invariant surface conditions and functional separation of variables
, 2000
"... This paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables. The relationship of these methods with nonclassical and weak point symmetries is stressed. Applications to diffusion equations w ..."
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Cited by 9 (0 self)
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This paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables. The relationship of these methods with nonclassical and weak point symmetries is stressed. Applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided.
Nonclassical And Conditional Symmetries
, 1996
"... features. Through the process of prolongation, which requires the group transformations preserve the intrinsic contact structure on the jet space, they define local groups of contact transformations on the kth order jet spaces J k . Such a transformation group will be a symmetry group of the sys ..."
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Cited by 9 (0 self)
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features. Through the process of prolongation, which requires the group transformations preserve the intrinsic contact structure on the jet space, they define local groups of contact transformations on the kth order jet spaces J k . Such a transformation group will be a symmetry group of the system of differential equations E ae J k if the transformations of the symmetry group leave E invariant. This implies that the group transformations map solutions of E onto solutions of E. The classical Lie symmetries are sometimes called external symmetries. To date, several extensions of the classical Lie approach have been proposed. Each of them relaxes one or more of the basic properties obeyed by classical symmetry groups. If we relax the restriction that the infinitesimal generators determine geometrical transformations on a finite order
Reduction Operators of Linear SecondOrder Parabolic Equations
, 2008
"... The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1 + 1)dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to ..."
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Cited by 8 (4 self)
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The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1 + 1)dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving the initial equations. The “nogo” result is extended to the investigation of point transformations (admissible transformations, equivalence transformations, Lie symmetries) and Lie reductions of the determining equations for the nonclassical symmetries. Transformations linearizing the determining equations are obtained in the general case and under different additional constraints. A nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented. An observed connection between reduction operators and Darboux transformations is discussed.
Singular Reduction Operators in Two Dimensions
, 2008
"... The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to firstorder ODEs are are exhaustively described. As examples, pr ..."
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Cited by 7 (3 self)
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The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to firstorder ODEs are are exhaustively described. As examples, properties of singular reduction operators of (1 + 1)dimensional evolution and wave equations are studied. It is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an indepth prior study of the corresponding singular vector fields.
Involution and Symmetry Reductions
 Math. Comp. Model
, 1995
"... After reviewing some notions of the formal theory of differential equations we discuss the completion of a given system to an involutive one. As applications to symmetry theory we study the effects of local solvability and of gauge symmetries, respectively. We consider nonclassical symmetry reducti ..."
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Cited by 7 (5 self)
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After reviewing some notions of the formal theory of differential equations we discuss the completion of a given system to an involutive one. As applications to symmetry theory we study the effects of local solvability and of gauge symmetries, respectively. We consider nonclassical symmetry reductions and more general reductions using differential constraints.